Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.17e5·2-s + 3.87e8·3-s + 3.77e11·4-s − 9.63e12·5-s + 2.77e14·6-s + 1.74e15·7-s + 1.71e17·8-s + 1.50e17·9-s − 6.91e18·10-s + 2.16e19·11-s + 1.46e20·12-s + 4.06e20·13-s + 1.25e21·14-s − 3.73e21·15-s + 7.14e22·16-s + 4.16e21·17-s + 1.07e23·18-s − 4.03e23·19-s − 3.63e24·20-s + 6.75e23·21-s + 1.55e25·22-s − 3.07e25·23-s + 6.65e25·24-s + 2.01e25·25-s + 2.91e26·26-s + 5.81e25·27-s + 6.57e26·28-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.74·4-s − 1.12·5-s + 1.11·6-s + 0.404·7-s + 3.37·8-s + 0.333·9-s − 2.18·10-s + 1.17·11-s + 1.58·12-s + 1.00·13-s + 0.783·14-s − 0.652·15-s + 3.78·16-s + 0.0718·17-s + 0.644·18-s − 0.889·19-s − 3.09·20-s + 0.233·21-s + 2.27·22-s − 1.97·23-s + 1.94·24-s + 0.276·25-s + 1.94·26-s + 0.192·27-s + 1.11·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $7.472397112$
$L(\frac12)$  $\approx$  $7.472397112$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3.87e8T \)
good2 \( 1 - 7.17e5T + 1.37e11T^{2} \)
5 \( 1 + 9.63e12T + 7.27e25T^{2} \)
7 \( 1 - 1.74e15T + 1.85e31T^{2} \)
11 \( 1 - 2.16e19T + 3.40e38T^{2} \)
13 \( 1 - 4.06e20T + 1.64e41T^{2} \)
17 \( 1 - 4.16e21T + 3.36e45T^{2} \)
19 \( 1 + 4.03e23T + 2.06e47T^{2} \)
23 \( 1 + 3.07e25T + 2.42e50T^{2} \)
29 \( 1 + 3.72e26T + 1.28e54T^{2} \)
31 \( 1 + 1.69e27T + 1.51e55T^{2} \)
37 \( 1 - 3.17e28T + 1.05e58T^{2} \)
41 \( 1 + 1.14e29T + 4.70e59T^{2} \)
43 \( 1 - 7.17e29T + 2.74e60T^{2} \)
47 \( 1 - 3.03e29T + 7.37e61T^{2} \)
53 \( 1 + 6.59e31T + 6.28e63T^{2} \)
59 \( 1 + 8.50e32T + 3.32e65T^{2} \)
61 \( 1 - 6.05e32T + 1.14e66T^{2} \)
67 \( 1 + 2.62e33T + 3.67e67T^{2} \)
71 \( 1 - 1.38e34T + 3.13e68T^{2} \)
73 \( 1 + 1.07e34T + 8.76e68T^{2} \)
79 \( 1 - 2.43e35T + 1.63e70T^{2} \)
83 \( 1 + 3.57e34T + 1.01e71T^{2} \)
89 \( 1 - 1.28e36T + 1.34e72T^{2} \)
97 \( 1 + 8.70e36T + 3.24e73T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.16494550432536679544830312999, −15.00510605312290870137908081243, −13.92877237768855401295678094908, −12.30695091169000167348543348909, −11.18580836267465425557109965037, −7.917206073367492893260757672301, −6.29938234778421846415242911055, −4.25974196820357533378371411840, −3.60206902088349860274362676812, −1.77757456369236737958588902134, 1.77757456369236737958588902134, 3.60206902088349860274362676812, 4.25974196820357533378371411840, 6.29938234778421846415242911055, 7.917206073367492893260757672301, 11.18580836267465425557109965037, 12.30695091169000167348543348909, 13.92877237768855401295678094908, 15.00510605312290870137908081243, 16.16494550432536679544830312999

Graph of the $Z$-function along the critical line